Initiating Change in Prospective Elementary School Teachers' Orientations to Mathematics Teaching by Building on Beliefs

REBECCA AMBROSE

INITIATING CHANGE IN PROSPECTIVE ELEMENTARY SCHOOLTEACHERS’ ORIENTATIONS TO MATHEMATICS TEACHING BY

BUILDING ON BELIEFS

ABSTRACT. Many mathematics educators have found that prospective elementaryschool teachers’ beliefs interfere with their learning of mathematics. Often teachereducators consider these beliefs to be wrong or naïve and seek to challenge them soprospective teachers will reject them for more generative beliefs. Because of the resili-ence of prospective teachers’ beliefs in response to these challenges, teacher educatorscould consider alternative ways of thinking about and addressing beliefs, particularly thepotential of building on rather than tearing down pre-existing beliefs. Data from an early-field experience linked to a mathematics-for-teachers course provide evidence that whenprospective teachers work intimately with children, in this case trying to teach 10-year-olds about fractions, the experience has the intensity from which beliefs can grow. Most ofthe prospective teachers in the study were surprised that mathematics teaching was moredifficult than they had anticipated. They began to consider the importance of providingchildren time to think when solving mathematical problems. The change described in thestudy is incremental rather than monumental, suggesting that building upon prospectiveteachers’ existing beliefs will be a gradual process.

KEY WORDS: field experience, mathematics pre-service teacher education, prospectiveteachers’ beliefs

INTRODUCTION

Prospective elementary school teachers come to their teacher educationprograms with a variety of beliefs that are influenced by their experi-ences as students in schools. Some believe that teaching will be relativelystraightforward, consisting primarily of offering clear explanations tochildren (Richardson, 1996). They believe that their abilities to relate tochildren and manage classrooms will be paramount to their success asteachers. Weinstein (1989) characterized this orientation as an optimisticbias, because prospective teachers enter their coursework assuming thatthey already know what they need to know in order to teach (see alsoFeiman-Nemser & Buchmann, 1986). These beliefs often lead prospectiveteachers to underestimate the complexity of teaching and the kind ofknowledge that they will need to be successful. In particular, they oftenunderestimate the importance of subject-matter knowledge in teaching.

Journal of Mathematics Teacher Education 7: 91–119, 2004.© 2004 Kluwer Academic Publishers. Printed in the Netherlands.

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Some mathematics education researchers have argued that teachers’subject-matter knowledge is extremely important and have documentedthe limited content knowledge of prospective teachers (Ball, 1990;Ma, 1999). Universities often require courses designed to enhanceprospective teachers’ mathematical knowledge by having them makesense of mathematics and understand the principles that underlie thearithmetic they memorized as children. Despite the promising designof these courses, prospective teachers’ beliefs about mathematics andteaching often diminish the outcomes of the courses. Some mathem-aticians have suggested that promoting changes in prospective teachers’beliefs about mathematics and mathematics teaching is critical to helpingthem to develop the content knowledge that they need to be effectiveteachers (Conference Board of Mathematical Sciences, 2001; Mathema-tical Sciences Education Board, 2001). But, teacher educators have beenmore successful in documenting the existence of beliefs that interfere withprospective teachers’ learning than in promoting belief change (Wideen,Mayer-Smith & Moon, 1998).

Often, efforts to change prospective teachers’ beliefs are initiated inmethods courses after subject matter courses have been completed andcome too late to support them in developing beliefs that will help them todevelop a deep understanding of fundamental mathematics. The researchreported here describes a program that attempted to initiate belief changeat the beginning of prospective teachers’ mathematical preparation. TheChildren’s Mathematical Thinking Experience (CMTE), part of the Inte-grating Mathematics and Pedagogy (IMAP) project, helped prospectiveteachers begin to understand the importance of subject matter knowledgein the teaching of mathematics by having them work with children in anelementary school while they were enrolled in their first mathematics-for-teachers course. For many of the prospective teachers, the field experiencewas one of their first experiences of working with children in schools and,for most, it was an intense experience that caused them to reconsider theirassumptions about mathematics teaching. I analyze the nature of the exper-ience, consider the factors that contributed to its intensity and examine theeffects of the experience on the prospective teachers’ beliefs. Specifically,it seems that the prospective teacher’s interest in relating to children provedto be a stimulus for expanding their views of teaching and affecting theirbeliefs about learning mathematics.

INITIATING CHANGE IN PROSPECTIVE TEACHERS’ BELIEFS 93

BELIEFS

Before describing the CMTE, I describe the framework I have adopted formy thinking about the process of belief change and the ways that teachereducators can promote belief change in prospective teachers. The compo-nents of this framework include several aspects of beliefs: their origins,their effects on one’s interpretations of experiences, the ways separatebeliefs combine to create belief systems, and the ways beliefs changewithin this framework. This framework provides support for the premisethat providing prospective teachers opportunities to work with children canbe a promising avenue to promote belief change.

Sources of Beliefs

Beliefs can be thought of as having one of two primary sources: emotion-packed experiences and cultural transmission (Pajares, 1992). The firstsource, emotion-packed experience, gives beliefs their “signature” quality.Many people can point to a vivid memory from which a particular beliefemerged (Nespor, 1987). For example, some prospective teachers givedetailed accounts of crying while they struggled to learn multiplicationtables. They relate these experiences to their belief that they are incapableof learning mathematics. The emotional component of these experiencesis one feature that differentiates beliefs from other forms of knowledge. Inrelating this feature to beliefs about teaching, Goodman (1988) suggestedthat these beliefs were derived from guiding images based on both positiveand negative experiences that teachers had as children.

The second source of beliefs, cultural transmission, creates beliefs thatmay be held at a subconscious level and can be thought of as resulting fromthe “hidden curricula” of our everyday lives. Culturally transmitted beliefsoften take the form of assumptions and stereotypes. For example, becauseprospective teachers’ mathematics work in school consisted mostly ofmemorizing procedures, many assume that mathematics always requiresmemorization, even though they have never heard a statement to that effect.People tend to be unaware of the culturally transmitted beliefs they hold,taking them for granted because they have neither examined nor discussedthem. These implicit beliefs may guide behavior in ways that could becharacterized as habits, with individuals doing things in particular waysthe reasons for which they are hardly cognizant.

Because of their origins, beliefs can be hard to change. It is impossibleto undo intense personal experiences or wipe out 20 years of livingin a culture. Teacher educators wishing to stimulate belief change in

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prospective teachers might pursue the avenue of creating for thememotion-packed experiences that will leave the vivid impressions thatform the basis of some beliefs. Another avenue of change is to develop acommunity that will instill positive implicit beliefs in prospective teachers.However, the duration of a teacher education program may be too short toachieve this kind of belief generation.

Effects of Beliefs

Beliefs have a filtering effect on one’s new experiences (Pajares, 1992).This filtering effect, which can make beliefs quite durable, is evidentwhen prospective teachers interpret experiences or information in theircourses in ways different from those their instructors intended (e.g., Simon,Tzur, Heinz, Kinzel & Smith, 2000). For example, one colleague hadher methods students work with kindergartners during the second weekof school. Her intention was for the prospective teachers to realize thatyoung children come to school with a great deal of informal mathematicalknowledge. After this field experience, the prospective teachers returnedto class impressed with how much the teacher had taught the children inthe first week of school (L. Clement, personal communication, January2001). Their beliefs that children’s knowledge of mathematics comesfrom formal school experiences led them to interpret the experience asevidence of teaching rather than of the children’s informal knowledge. Thispowerful filtering effect of beliefs is responsible for their important role inlearning and leads to teacher educators’ endeavors to affect the beliefs ofprospective teachers.

Belief Systems

Beliefs, whatever their source, are related to one another, forming systemsin which related beliefs are connected (Rokeach, 1968). Green (1971)pointed out that belief clusters might be held in isolation, unconnectedto other belief clusters. He wrote, “We tend to order our beliefs in littleclusters encrusted about, as it were, with a protective shield that preventsany cross fertilization among them or any confrontation between them”(p. 47). For example, some prospective teachers believe that childrenshould have opportunities to be creative. This belief might be connectedto other beliefs about art and writing and may come from childhood exper-iences in these domains. The belief about the importance of creativity forlearning may not be connected to beliefs about mathematics because theprospective teachers have not had creative experiences in mathematics.Teacher educators could help prospective teachers connect their beliefabout the importance of creativity for learning with their beliefs about


learning of mathematics so that they begin to look for the creative poten-tial in problem solving or problem posing. Green (1971) suggested thathelping people to develop well-connected belief systems should be one ofthe primary purposes of teaching.

Rokeach (1968) argued that beliefs related to a particular situation orobject, form attitudes similar to the belief clusters that Green (1971) iden-tified. For example, prospective teachers’ beliefs about teaching wouldtogether form an attitude about teaching. Rokeach pointed out that onedimension of attitudes is their degree of differentiation. Well-differentiatedattitudes are those that have a large number of parts and are well articu-lated. Attitudes formed by culturally transmitted beliefs can be undifferen-tiated and fairly simplistic. These might be in the form of “teachers shouldbe nice” or “teachers should make class interesting”. These undifferenti-ated attitudes do not encompass the complexity of the situations to whichthey apply and are similar to stereotypes in that the believer assumes thatthey can treat all situations of this type as if they were the same, instead oftaking into account the particulars of the situation.

Undifferentiated attitudes are often ones that have not been examined.Fenstermacher (1979) pointed out that the role of teacher educationprograms should be to support teachers in bringing tacit beliefs into theopen so that these beliefs can be transformed into objectively reasonablebeliefs. This process helps prospective teachers to make more principleddecisions on the basis of beliefs they believe are important, instead ofacting on the basis of the habits of unexamined beliefs and undifferentiatedattitudes. At the beginning of their course work, prospective teachers tendto have undifferentiated attitudes about teaching because they have not hadchances to refine them through the reflection that Fenstermacher discussedand may be one of the reasons that prospective teachers undervalue theirsubject matter preparation.

Changing Belief Systems

Four mechanisms for stimulating belief-system change in prospectiveteachers have been outlined above: (a) they can have emotion-packed,vivid experiences that leave an impression; (b) they can become immersedin a community such that they become enculturated into new beliefsthrough cultural transmission; (c) they can reflect on their beliefs so thathidden beliefs become overt; (d) they can have experiences or reflectionsthat help them to connect beliefs to one another and, thus, to develop moreelaborated attitudes. Many teacher education programs rely on reflection asa means for fostering belief change. Stofflett and Stoddard (1992) pointedout that this practice can serve to make prospective teachers more articulate

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and definite about the beliefs that they held before their teacher preparationstarted, but may not serve to help them form new beliefs. Vivid experiencescoupled with reflection may be required for new beliefs to form.

A fifth kind of belief change, the most dramatic, is the reversal ofexisting beliefs. A prospective teacher may change, for example, frombelieving that new material is best presented through a teacher’s lecturingto believing that new material is best developed by students’ grapplingcollectively with the material. This kind of conversion is the goal ofmany teacher educators and, all too often, they are disappointed whenprospective teachers fail to reverse their existing beliefs. McDiarmid, Balland Anderson (1989) found that the prospective teachers in their studyfailed to reverse their beliefs about teaching as telling. They found that theprospective teachers “expanded the range of options for teaching mathe-matics” (Wideen et al., 1998, p. 344) but were not converted to a wholenew way of thinking about mathematics teaching.

Prospective Teachers Beliefs About the Nature of Teaching

To build on prospective teachers’ beliefs, one must recognize that theirbeliefs about mathematics teaching and learning are part of a larger systemof beliefs that also includes beliefs about teaching, generally. For example,two central beliefs of many prospective teachers are that teachers shouldbe nice and should present instruction clearly.

Most elementary school teachers choose to enter the profession becausethey care about children (Hargreaves, 1994; Howes, 2002; McLaughlin,1991) and place strong emphasis on affective and interpersonal issues(Weinstein, 1990). Gellert (2000) found that the prospective teachers inhis study preferred to shelter students from challenging mathematicalproblems to protect them from anxiety. He attributed this attitude to thetendency of prospective teachers to see themselves in the role of nurturers.Beliefs about the importance of relationship building endure into the firstyears of teaching when new teachers often expend a great deal of energytrying to find ways to develop positive relationships with their students(Hollingsworth, 1992). In reviewing the research literature on learning toteach, Wideen et al. (1998) concluded, “Beginning teachers value socialand peer groups, positive self-concept, and helping behaviors” (p. 142).The centrality of prospective teachers’ beliefs about caring indicate thatthey will value experiences in which they can be intimately involved withchildren.

In addition to believing that nurturing is one of the primary func-tions of teachers, many prospective teachers believe that teaching entailspresenting information that student will memorize (Richardson, 1996).


They assume that explaining is a fairly straightforward enterprise (Feiman-Nemser, McDiarmid, Melnick & Parker, 1988). “Teaching itself is seenby beginning teachers as the simple and rather mechanical transfer ofinformation” (Wideen et al., 1998, p. 143). This is another undifferenti-ated attitude that eventually becomes elaborated as prospective teachersgain teaching experience. Unfortunately, as Weinstein (1989) discovered,this undifferentiated view of teaching leads prospective teachers to under-estimate the importance of their subject-matter preparation. Typically,only after they have finished their subject-matter course work and havehad teaching experience, do prospective teachers begin to recognize thecomplexity of teaching and the kind of knowledge required to do it well.

Prospective teachers’ views of ‘teaching as caring’ and ‘teaching asexplaining’ form the basis of their belief systems and will be likely to beretained while their belief systems change and develop. We hypothesizethat prospective teachers’ beliefs about children and the nature of teachingare more central to their belief systems than their beliefs about mathe-matics. Our goal in the IMAP project was to take advantage of prospectiveteachers’ interest in children. Others have found that the personal relation-ships that emerge from individual tutoring sessions have positively affectedprospective teachers’ learning in the domain of reading instruction (Worthy& Patterson, 2001). We hoped that while they worked with children inthe domain of doing mathematics, the prospective teachers’ interest intheir students would motivate them to expand their view of teaching andof mathematics when they encountered the limitations of an exclusive‘teaching as explaining’ approach. We saw their interest in children as thevehicle for motivating them to care about mathematics and mathematicsteaching and to begin to alter their views. We also expected that whenthey had experienced some teaching, their undifferentiated attitudes aboutteaching as telling would become elaborated.

Generative Beliefs for Learning Mathematics

For our project we identified several beliefs that we hoped the prospectiveteachers would develop. One is the belief that mathematics is a webof interrelated concepts and procedures. Related to this view are beliefsabout the relationship between concepts and procedures: that knowledge ofconcepts is more powerful and generative than knowledge of proceduresand that one can know procedures without understanding the underlyingconcepts. If prospective teachers begin to appreciate the importance ofconcepts in developing mathematical understanding, they might try todevelop their own conceptual understanding as well as think of ways toteach for conceptual understanding. Other beliefs are related to teaching

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and learning mathematics and include (a) believing that children bringto school a great deal of informal mathematical knowledge that can bethe basis of instruction and (b) recognizing that often the ways childrenthink about mathematics differ from the ways of adults who have beenschooled.

METHOD

Description of the Early Field Experience and the Mathematics Course

Fifteen prospective teachers volunteered to participate in the CMTE, anexperimental course, which required that they enroll in both the CMTE andtheir mathematics course. They were compensated for their participation.At the large urban regional university where the CMTE was offered, therewere twelve sections of the mathematics-for-teachers course taught by sixdifferent instructors. All instructors used the same text and the same finalexamination. The instructor of the CMTE also taught the mathematicscourse for the fifteen prospective teachers so that he could integrate thetwo experiences as much as possible. He was a professor with a researchinterest in children’s mathematical thinking. The CMTE met eight timesfor a two hour period at a local elementary school.

The prospective teachers explored number and operations in both themathematics course and the CMTE. The vision of the course was alignedwith that described by the Conference Board of Mathematical Sciences(2001), which emphasized the importance of prospective teachers’ appre-ciation for the intellectual richness of elementary school mathematics.The goals for the course were for prospective teachers (a) to make senseof “non-standard methods commonly created by students, the reasoningbehind the procedures, and how the structure of number is used inthese calculations” (Conference Board of Mathematical Sciences, 2001,Chapter 3); (b) to understand the variety of ways that the operations canbe interpreted; and (c) to be able to represent mathematical concepts in avariety of ways. The domain of the course was whole and rational numbersand included analyses of children’s invented approaches to problems.

In the CMTE, pairs of prospective teachers worked with individualchildren using specific tasks and activities designed to elicit children’sthinking; the emphasis was on problem solving rather than on symbolmanipulation. Each prospective teacher worked with a partner; one in eachpair led the problem-solving session, and the other took notes. Each partnerhad a chance to perform each role several times. The partners were encour-aged to help each other. They often exchanged ideas about which problem


to present next, what question to ask, and so on. The partners discussed theexperience afterward and considered issues that arose during the session.

During the first weeks of the course, the prospective teachers had threesessions with 6 to 9-year-old children. The goal in this phase of the CMTEwas to influence the prospective teachers’ beliefs about children’s informalknowledge and the children’s tendencies to act out story problems. Theprospective teachers provided the children with various whole numberstory problems that could be modeled using each of the four operations.For the last four weeks of the CMTE, each pair of prospective teachersworked with one 10-year-old child on fraction concepts. This part of theCMTE was designed to show that a concrete approach to this difficultconcept can help children develop understanding. The prospective teachersworked with the same child during these sessions so that they could try toteach the child over a period of time.

Participants and Site

Of the 15 prospective teachers in the CMTE, 13 were female and twowere male; five were in their first year of university course work; eightwere in their third year of university course work (six of whom had trans-ferred from 2-year colleges); and two were postgraduates completing themathematics prerequisite for the teaching-credential program. Three ofthe prospective teachers spoke English as their second language; sevenof the prospective teachers reported that they had enjoyed mathematics aschildren and felt fairly successful with it. The other eight reported thatthey found mathematics to be boring and difficult to learn when they werechildren.

The prospective teachers worked with children at a multiethnic, urbanelementary school in which 46% of the students were White, 39% wereHispanic, 10% were African American, and the remaining students repre-sented a variety of other ethnic groups. Many of the children with whomthe prospective teachers worked were bilingual. The school used a drill-based mathematics curriculum, in which emphasis was on acquisition ofstandard procedures; independent seatwork was the predominant mode ofinstruction.

Data Sources and Analysis

Data for the study came from a variety of sources including surveys,interviews, prospective teachers’ written work, and field notes. Each pro-spective teacher completed a computerized belief survey at the beginningand end of the CMTE. The survey was a pilot version of a belief surveythat was later used in a large-scale study of the CMTE (see Ambrose, 2002,

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for more information about the survey). The survey consisted of six open-ended items focused on whole number and rational number arithmetic. Tovalidate the inferences we made from the belief survey, we asked eachprospective teacher to discuss and elaborate on the responses given onthe survey that they had just completed. Two survey items were of partic-ular importance for this study. Both required the prospective teachers toconsider alternatives to standard algorithms, one in the domain of multi-digit addition and another in the domain of multidigit subtraction. Theseitems were used to determine the degree to which the prospective teachersbelieved in the importance of multiple approaches.

Each prospective teacher was interviewed at the beginning and endof the semester and the interviews were transcribed. In the initial inter-view, interviewers followed a protocol that included questions about theprospective teachers’ attitudes toward mathematics, their thoughts aboutteaching and learning mathematics, and follow-up questions about thebelief survey. The interviews were audiotaped and later transcribed. Astaff of four researchers collected field notes of the prospective teachers’problem-solving sessions with the children. Audiotapes of the problem-solving sessions were used to augment the field notes. Each prospectiveteacher was videotaped once during the semester while working with achild. After each problem-solving session, the prospective teachers wroteshort personal-reaction papers (Quickwrites) in which they shared theirinitial impressions; they wrote longer reflections as homework. Thesereflections were collected and photocopied before being returned to theprospective teachers.

My data analysis was an emergent process similar to that used ingrounded theory (Glaser, 1998) and began with intense analysis of thedata of one prospective teacher (Donna) in the group. This analysis wason-going during the semester when the CMTE was held. I read and rereadDonna’s data, looking for emergent themes, in particular, for what aspectsof the CMTE she found compelling and how these experiences affectedher beliefs. I developed a set of codes and used them to analyze data fromfour more prospective teachers to confirm or contradict hypotheses. Fromthis analysis, I saw that the sessions with the 10-year-olds had the greatesteffect, and some factors related to that experience emerged as being crit-ical to the prospective teachers’ belief change. I then analyzed those datarelating to the work with the 10-year-olds from the other 10 prospectiveteachers and coded it according to the nature of the problem-solvingsessions, the cognitive demands the sessions placed on the prospectiveteachers, and the emotional aspect of the work. In charts, I organizedsegments of coded data from the whole group of prospective teachers to


determine the extent to which various factors affected them and to onceagain confirm or contradict hypotheses.

Pajares (1992) pointed out that beliefs must be inferred; they cannot bedirectly measured. In assessing the prospective teachers’ beliefs, I lookedfor their statements and actions that indicated the beliefs they held. Basedon the same information, others might come to different conclusions aboutthe prospective teachers’ beliefs, so I offer verbatim quotes on which otherscan base their conclusions about the beliefs of the prospective teachersin the study. Characterizing the beliefs of a group of individuals can beproblematic because individuals in the group can hold different beliefs.In several cases, I discuss tendencies within the group and do not meanto imply that all the prospective teachers in the group developed identicalbelief systems. Because of the sometimes hidden nature of beliefs, eventhe individuals who hold them may be unaware of their presence. In thissense, any analysis of belief change will require interpretation, and I offermy interpretation with the recognition that there is a subjective componentto the interpretation.

THE NATURE OF THE CHILDREN’S MATHEMATICALTHINKING EXPERIENCE

Intense Teaching Experiences

Before considering the belief change that emerged from the CMTE, I beginby establishing that the prospective teachers found the experience to beintense. In their final interviews, when asked about a CMTE episode thatstood out for them, 12 of the 15 prospective teachers discussed their workwith the 10-year-olds as being the most memorable. Five discussed theexcitement they felt when their student told them he or she had learnedsomething. Julie commented, “I was so impressed that he remembered.I was just excited that I actually made an impact”. Five discussed theirconcerns that their student struggled with concepts. Holly said, “I wasn’texpecting it to be such a slow process, like with fractions. I found that shewas way behind what I thought she could understand”. Lisa spoke abouther flawed assumptions about her 10-year-old student’s understandingof English, and Donna spoke about the power of a real-world contextto support her 10-year-old student’s thinking. At some point in theirinterviews, all the prospective teachers talked about their work with the10-year-olds as being important learning experiences for them.

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The focus of the four sessions with the 10-year-old children was frac-tions. After an assessment session, the children explored, using the patternblocks, relative sizes of fractions and different names for fractions greaterthan 1. The children also solved equal-sharing problems that resultedin mixed-number answers. The tasks were designed to build conceptualunderstanding and to reduce the emphasis on the symbolic work that canlead children to misconceptions (Mack, 1995). The prospective teacherswere encouraged to make instructional decisions, during the sessions, toadapt the work to each child’s level of understanding. Some prospectiveteachers introduced the children to adding fractions whereas others spentmore time on fractions greater than 1.

In the first fractions session, the prospective teachers found that thechildren were relatively unfamiliar with fractions. The children did nothave a feel for the size of 12/13. They struggled to compare fractions.For example, many thought that 1 was greater than 4/4. The children werenot familiar with converting improper fractions to mixed numbers. Mostclaimed never to have seen improper fractions and did not know howto interpret them. They could partition wholes into parts but had troublenaming the parts they drew.

The second fraction session, which involved using pattern blocks, wasa high point for most of the CMTE pairs. The children enjoyed workingwith the pattern blocks to build representations for a variety of fractionalquantities and were successful using the pattern blocks to compare simplefractions such as 1/3 and 1/2. Many of the prospective teachers conveyedthis enthusiasm in their writing about the session. For example, Phanwrote, “Our child was on a roll. She would laugh out loud each time wegave her a fraction number”. In reflecting on this experience, Lisa wrote,

I was amazed by the progress he made in such a short amount of time. In the end hewas able to push the blocks aside and picture them [fractional quantities] in his mind . . . .Afterwards I felt very proud because I think we honestly helped him with his understandingof fractions.

By the end of the session all the children had converted some improperfractions to mixed numbers without the aid of the blocks, and most of theprospective teachers were excited by the progress the children had made.

The third fraction session surprised most of the prospective teachers.During a group discussion prior to the session, they decided to givethe children some problems identical to those they had solved in theprevious session: converting improper fractions into mixed numbers. Theprospective teachers elected to present the problems in symbolic formwithout giving the children manipulatives. They asked the children toconvert, for example, 7/6 (written in symbols) into a mixed number, and


they found that the children struggled with such review problems. Onechild claimed that 8/5 and 5/8 were the same, and another child haddifficulty drawing a representation for 3/2.

The contrast between the second and third fraction sessions impressed13 of the 15 prospective teachers. The passages in Table I are representa-tive of the reactions of the prospective teachers to their work in the twosessions. Each of the six prospective teachers quoted in the table workedwith a different child.

TABLE I

Prospective Teacher’s Reactions to Their Work with 10-Year-Olds

Prospectiveteacher

Comments following fractionsession 2 (pattern-block session)

Comments following fractionsession 3

Jane I now have the greatest feelingbecause I feel that our studentreally progressed. I feel like wereally taught him something and heunderstood.

He seemed to forget some of themore basic concepts of fractions.I was disappointed in some of hisresponses to the easier questions.

Kathy This week was so impressive. Ourstudent has improved so much overthe past week.

Today was so shocking. Last timeI walked away saying “WOW”. Itseems to me that he had learned somuch in just a week. Then today itwas like we took a backwards stepand he had forgotten everything. . . . I was very humbled today.

Ana I am amazed by the progress hemade just by using the manipu-latives. I really feel like he hasa better understanding of fractionsthan he had last time when wewere just focusing on the writtensymbols.

In an effort to make our studentcomfortable and relaxed, wedecided to begin our interviewwith easy review questions. Weasked him to draw one and a half.Pretty simple. He couldn’t do it. Ifigured that writing the numbersdown would help. It didn’t. Weassumed these tasks would beeffortless for him because heseemed to understand them soeasily in our last interview.

Gloria Wow, I am really impressed. Sheseemed to have improved since thelast time we interviewed her. Mypartner did a great job of teaching

It seemed like our student didn’tretain anything. She made the samemistake and it took her just as logto draw the fractions and compare

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TABLE I

Continued

Prospectiveteacher

Comments following fractionsession 2 (pattern-block session)

Comments following fractionsession 3

her that 11/4 is the same as 2 3/4.That was pretty intense.

8/5 and 5/8 . . . . It was a frus-trating session because it seemedlike we went backwards . . . . Thissession was so aggravating. She didnot seem to retain anything we hadworked . . . nothing seemed to sinkin.

Phan Overall the child improved a lotsince last week . . . . I think shewill learn from this experience andremember it forever.

This interview was shocking andyet very sad. She could notremember how to do just simplefractions. This had disappointed usbecause we thought she had nailedfraction problems last time we met.I learned I cannot teach a child yet.

Donna Our session went surprisingly well.I was so stoked as we taught herhow to do mixed numbers andimproper fractions and she pickedup on it and was able to write herown. She even was able to do 23/12into 1 11/12 and 10/4 into 2 2/4. Iwas amazed.

Our student had forgotten much ofwhat was taught her from the lasttime . . . . She was more confusedthat anything else today.

The terms stoked and excited used by the prospective teachers todescribe the pattern-blocks session point to the intensity of the experience.The prospective teachers’ initial reactions reflect the optimistic bias thatWeinstein (1989) identified and might strike the reader as naïve or exag-gerated. Keep in mind, however, that some of the prospective teachers were19 years old and had limited experiences working with children. Theiroriginal assumptions that they had been successful were based on whatsome might argue was a relatively limited period of time with a narrowrange of fractions (those that could be represented with the pattern blocks).Several of the prospective teachers’ comments indicated that they thoughtthat the teacher was responsible for the success of the session: “We reallytaught him something”; “My partner did a great job of teaching”.

“Today was so shocking.” “This had disappointed us.” “So aggrav-ating.” These reactions to the follow-up session express the prospective


teachers’ chagrin and surprise at their students’ lack of retention. Theirsurprise reflects the novelty of this experience for them. Experiencedteachers would be unlikely to be surprised that the children had troubleremembering a symbolic approach to converting fractions after beingexposed to it for only a short amount of time, but for these novice teachers,this result was unexpected. The experience was particularly troubling tothe prospective teachers because they felt that the children had been sosuccessful during the pattern-block activity, and many felt responsible forthat success. Their comments (“I was very humbled today”; “I learned Icannot teach a child yet”) show how personally the prospective teacherstook this experience. The emotional charge of the experience contributedto its intensity and left the kind of vivid impression from which beliefs cangrow.

Two of the prospective teachers were less affected than the others bythese two sessions and were cautious in their evaluations of what thechildren understood from the pattern-block session. Nina noted that herchild had been successful, but she was not convinced that the child’sunderstanding was completely developed. She wrote, “I was impressedby my student’s advancement from the week before. She came a longway. However, I don’t feel like she really understands what she is doingfully”. Holly was concerned by her child’s belief that there are no frac-tions in which the numerator is greater than the denominator. She wrote,“I was frustrated. I could see Stephanie’s confusion, but I don’t know ifI parted the clouds for her”. These two prospective teachers seemed notto have felt the elation or let down as intensely as their peers, and thesessions may not have engendered new beliefs for them. Their cautionin reacting to the pattern block session indicates the reflective stance thatthese two prospective teachers had throughout the CMTE. For these partic-ular teachers, the belief-system change that they exhibited may have beendue more to their ongoing reflection than to an intense experience.

Several features were in place that created the conditions necessaryfor these teaching experiences to have the intensity that they did for themajority of the prospective teachers. These features will be explored in thenext section.

Focus on Mathematics Learning Rather Than on Classroom Management

The fact that prospective teachers worked with individual children contrib-uted to the intensity of the experience. Because they were working one-on-one with the child, the prospective teachers did not face the cognitiveoverload that can accompany teaching. When prospective teachers arein student-teaching situations, they must attend to all the issues of class

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management and content, and their cognitive capacity is overwhelmedby all the stimuli they are trying to assimilate (Hollingsworth, 1989).Through their work with an individual child, the prospective teachers hadclear evidence that the child was struggling and that their teaching hadnot been entirely successful. They could concentrate their thinking on thisissue because they were not distracted by the host of management issuesthat typically preoccupy student teachers. They could not attribute thechildren’s difficulties to their behavior, attention span, or attendance. Nordid they have the option to turn to a different child to get the answers theysought; no other children could “bail out” the children or the prospectiveteachers. They had to face the fact that this was difficult material to teachand to learn.

High Cognitive Demand

Although the CMTE was stripped of some factors that occupy teachers’minds, the prospective teachers had to think about several things while theyworked with their children. They experienced what some call “knowledgein use” (e.g., Ball, 2000), that is, the knowledge that teachers must usewhile teaching. They had to consider the mathematical concept at hand,attend to what the child was doing and consider what understanding thechild had. They had to decide what question to ask or what problem toprovide in order to extend the child’s understanding and what representa-tion might help the child better understand the concept. Lisa talked aboutthe challenge of this cognitive demand, observing “trying to work withthem and think on my toes and figure out what questions to ask really fastwas hard”.

Donna talked about the difficulty of finding an appropriate vocabularythat would make sense to the child:

Donna: Sometimes you need to change your vocabulary or whateverso that it fits their world. Sometimes you don’t know whatto say. You feel like you understand – like you’re explainingthe right thing.

Interviewer: It makes perfect sense to you.Donna: Yeah. but not to the child. Sometimes you can’t explain

things.

Many prospective teachers talked about developing explanations as one ofthe most challenging aspects of their work with the children. Julie said,

I had no clue how to explain this one problem. He was just looking at me like, “Explainit”. I was just like, “I don’t know!” I didn’t even know how to explain it . . . . I know how


to find the common denominator for an addition problem, but I didn’t know how to teachit. So that was what was hard.

Kathy said, “I was trying to explain it to him, and I confused myself. It washorrible”. When they struggled to find the words to generate clear explan-ations, the prospective teachers experienced the high cognitive demand ofteaching. This was a novel experience for most of them, because few hadever been in the position of having to think on their feet in this way. Thecognitive demands of teaching coupled with the novelty of this kind ofthinking contributed to the intensity of the experience.

Connecting with Children through Mathematics

One critical feature of the work in the CMTE was the interpersonalcomponent. Working with the children on fraction concepts provided theprospective teachers opportunities to connect with children and to developrelationships, a feature evident in some prospective teachers’ commentsabout their children. Julie wrote, after her last session, “I’m so sad that Iwon’t be seeing him anymore. He had to have been the most polite child Ihave ever seen. He was so sweet . . . . I miss him already”. Although thiscomment may seem overly sentimental, the prospective teacher writing itwas one of the 19-year-old freshmen who had limited experiences workingwith children. These were the first words she wrote when reflecting on herfinal problem session, showing that this personal relationship was the mostsalient aspect of the work for her. Joe brought a present for his child to thelast interview and mentioned how much he had enjoyed getting to knowher. Tom and Alison commented that they were touched that their childcared so much about their work with her that she held on to a mathematicspaper for two weeks and brought it back to one of their sessions.

The one-on-one teaching situation was intimate in that the child wasasked to share his or her thinking, and the prospective teachers werecommitted to listening. Goldstein (1999) wrote about this kind of inter-action as involving both an intellectual component and an emotionalcomponent, requiring engagement and receptivity on the part of theteacher. As Tom noted,

Going and dealing with the student really kept me on my toes, because we had that addedresponsibility. I wasn’t just responsible for my own time and knowledge, but I was actuallygoing to be meddling with somebody else’s knowledge and time. I think that made mefocus more.

Kathy noted how important working with a child was for her: “Where elsedo you get an opportunity to sit with a child and have the child share whatshe is thinking? It was almost an honor to have the child do that. I felt

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very privileged”. The prospective teachers developed emotional ties thatcontributed to their learning because of their personal investment, which,along with the focus on the mathematics learning of one child and the highcognitive demands of this work, made the CMTE the kind of engrossingexperience with the emotional charge that leaves an impression which cangive rise to beliefs.

FRACTIONS WORK’S EFFECTS ON PROSPECTIVETEACHERS’ BELIEFS

Beliefs about Teaching Mathematics

The fractions-teaching experience affected the ways that many of theprospective teachers viewed teaching mathematics. They began to recog-nize that teaching requires more than simply presenting information tostudents. At the end of the semester, Donna commented about what shewould tell other prospective teachers: “Well, like I said before, not toexpect that a child knows what you’ve taught ‘em, because just becauseyou’ve taught ‘em doesn’t mean that they understand it”. Although Donnamay not be using the term understanding in a conceptual sense and mayinstead be talking about a student’s ability to remember a procedure,her comments reflect her recognition that teaching does not equate withstudent learning. We inferred, from the way she phrased her response, thatbefore the CMTE this prospective teacher had expected that “a child wouldknow what you’ve taught ‘em”, reflecting an initial stereotypical attitudethat teaching simply entailed presenting information to students.

Tom commented about how his views of teaching had changed: “Youthink, ‘Oh well, I’ll just tell them this and they’ll understand it’. And thenwhen you work with kids, you realize that it doesn’t work that way”.Tom’s transmission view of teaching had been expanded when he real-ized that children did not readily learn the material that he thought he hadtransmitted to them. Kathy stated,

I went into class that day thinking, “I’m so excited. I’m going to teach him this. By the endof the hour, he’s going to know it and he’ll be able to do it forever”. And it didn’t happenthat way, so I guess to just keep that in mind and to know that it’s not going to only take anhour for a child to understand a concept.

Kathy originally assumed that her student would absorb and retain theinformation she presented, and, through the CMTE, she learned thatteaching was not as straight forward as she had initially thought.

After their experiences in the CMTE, most of the prospective teacherstalked about the importance of providing children time to think, both


during individual problem-solving sessions and over long periods of time.Donna replied, in responding to the question “What did you learn from theCMTE?”

. . . not drill it into their heads. When children learn, they need their own space and time tolearn on their own. Let them have a chance first, and then see what they need help with.

We inferred from Donna’s suggestion, “Let them have a chance first andthen see what they need help with”, that she was beginning to developa more student-centered perspective toward teaching that might includea teaching-as-telling orientation but also included a concern for carefullytiming her lectures and assessing what the children knew first. Her morestudent-centered perspective was reflected when she later said,

I wanted to help her, but I also wanted her to do it on her own. I didn’t want to step in toomuch . . . . I have learned to give my child enough time on her own to try and figure out aproblem before I jump in and help her.

These comments illustrate how Donna’s attitude toward teaching becamedifferentiated. She began with a stereotypical and simplistic attitude,consisting of the belief that teaching entails presenting information. Shecontinued to hold this belief, as was evident in her comments aboutwanting to help the student. She added to that belief another belief aboutteaching as stepping back to allow students time to think. Her attitudetoward teaching grew to include two beliefs that were connected, and inthis way the attitude became differentiated.

The issue of providing children with time was a recurring theme inmany of the prospective teachers’ suggestions for people who might parti-cipate in the CMTE. Kathy mentioned, “Give them time – don’t bombardthem with questions”. Jane stated,

They don’t really learn anything when you just give them the answer. You just have to givethem time. You can’t just push them and keep asking questions, because when I did that,they were just frustrated.

Julie said, “Give them time and don’t show them everything. You have tolet them discover it for themselves, but you can help them along”.

Jane and Julie equated providing children time to think with allowingthe children to figure things out for themselves. They cautioned against“giving answers” and “showing them everything”, indicating that theyhad expanded their attitude toward teaching beyond merely presentinginformation to include facilitating children’s thinking. Nina explained herexpanded view:

Teaching is not me giving the information, and then them absorbing it, but rather givingthem the tools that they need to learn on their own. I think that’s probably the mostimportant thing that I learned.

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The prospective teachers characterized the idea of giving children time tothink and letting them discover things for themselves as insights that theyhad gained from the CMTE. Apparently they had not started the semesterwith these ideas but developed them while working with the children. Wetook this as evidence that their attitude about teaching expanded as a resultof their experience.

All the prospective teachers recognized that teaching is not as simpleas they had expected it to be. Holly summarized her learning:

It was a lot more complex than I expected. It was also good talking to all my classmates,seeing that it’s not as clean-cut as we thought it would be.

Most came to believe that providing children with “think time” was animportant element to good teaching. Several came to believe that childrenshould have opportunities to figure things out for themselves, and afew developed faith that children could learn on their own when givenappropriate tools.

Beliefs about Multiple Solution Strategies

The prospective teachers grew to appreciate the importance of multiplesolution strategies in mathematics. Their appreciation was apparent in theircomments in interviews as well as in their belief-survey responses but wasless apparent in their work in the problem-solving sessions.

In their interviews, the prospective teachers focused on the importanceof knowing different mathematical approaches for successful teaching. Inher final interview, when asked what she had learned from the CMTE,Donna stated that she needed “to be able to know how to do things morethan just your way – that your way doesn’t work for everybody. Kidslearn in different ways”. Nina noted the importance of being flexible: “Youhave to be able to be flexible and approach math problems from differentperspectives. Each child’s learning is going to be different”. For theseprospective teachers, being familiar with different approaches would allowthem to assist different children.

Tom noted that being familiar with different approaches to problemsolving helped him in his teaching. He mentioned “getting away fromformulas. Using manipulatives and drawing pictures seemed to really helpthem make sense of what was going on”. These comments indicate aninterest in having all children use manipulatives and pictures as a wayto deepen their understanding of the concepts. He was not advocatingmultiple strategies to meet multiple needs; instead, he was arguing infavor of individuals knowing multiple strategies as a means for makingsense of the mathematics. Responses to the belief survey provided further


evidence for changes in the prospective teachers’ beliefs about multiplesolution strategies. In considering the addition segment (see Figure 1), allthe prospective teachers wanted more strategies shared at the end of thesemester than they had at the beginning of the semester.

Figure 1.

Six of the prospective teachers who originally wanted one or twostrategies shared wanted four or five strategies shared at the end of the

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semester. At the beginning of the semester, two prospective teachers notedthe connection between the addition strategies and place-value under-standing whereas at the end of the semester, eight commented on theimportance of helping children make this connection. By the end of thesemester all the prospective teachers believed in the importance of multiplesolution strategies. For some this belief was evident at the beginning of thesemester but seems to have been strengthened; for others it had developedover the course of the semester. All the prospective teachers saw value inmultiple solution strategies, and some saw multiple solution strategies as avehicle for promoting conceptual understanding.

The prospective teachers were asked, in another segment on the beliefsurvey, which of two strategies they would prefer their students use (seeFigure 2). At the beginning of the semester, nine preferred that studentsin their classrooms exclusively use the standard algorithm for multidigitsubtraction. In supporting this choice, many wrote that it is less prone toerror, faster or “more of a sure thing”. We inferred from these responsesthat these prospective teachers were focused on the production of answers.At the end of the semester, 13 wanted their students to have access to boththe standard algorithm and an alternative approach. (Of the two remaining,one wanted her students to use the standard algorithm, and the otherwanted her students to use the alternative approach.) In explaining whythey wanted children to have access to both strategies, five mentionedthat children learn in different ways and so should have a choice ofstrategies. We inferred from their responses that although these prospectiveteachers were open to multiple solution strategies, they also believed thatcomputing correct answers is a focal point for instruction. In contrast, sixmentioned that children should learn both ways so that they would betterunderstand the concepts associated with the procedures. We inferred thatthese prospective teachers had become interested in having their studentsdevelop conceptual understanding. Julie wrote, “I would want them tounderstand the concepts behind the traditional way”. This response wasin contrast to Alison’s: “I think both are important because some kidsmay be able to solve the problem easier with the other way”; her focusis on generating correct answers rather than on understanding concepts.As was evident in the interview data, some prospective teachers wereinterested in multiple strategies to accommodate different learning styles,whereas others were interested in multiple strategies as a means to promoteunderstanding.

Although the prospective teachers talked about valuing multiple solu-tion strategies, this belief was not always evident in their work with the10-year-olds. When they had opportunities to make instructional decisions,


Figure 2.

three of the seven groups chose to teach their students the symbolicprocedure for adding fractions by finding common denominators. Whenthey did so, they neither supported their students in developing their ownapproaches for solving these problems nor presented multiple strategies.They attempted to explain the procedure for finding common denominatorsand in so doing ran into difficulties. Margie explained her experience: “Itotally confused her so much . . . . I didn’t even think of showing her withthe blocks. I was just showing her the procedure way. Then afterwards Ithought, ‘Oh God, I should have showed her this way’ ”. We were some-

114 REBECCA AMBROSE

what disappointed when we saw the prospective teachers focusing on thestandard symbolic procedure after they had discussed the value of multipleapproaches. We were relieved that at least Margie recognized that shemight have approached her instruction differently.

Beliefs about the Importance of Mathematics Understanding for Teaching

The prospective teachers’ experiences in the CMTE helped them to realizethat their understanding of mathematical concepts was essential to theirsuccess as teachers. Cindy stated,

I want to teach young children, so I didn’t think I needed to know a whole lot of actualmathematical skills and I really disagree with that now. In order to come up with a creativeway to teach it, you need to understand what you’re talking about and you need to have themath skills to do that.

This prospective teacher came to the realization that even mathematicsfor young children was complicated to teach and required conceptualunderstanding.

Many of the prospective teachers talked about how the CMTE helpedthem to appreciate the importance of the material that they were learning intheir mathematics course. Margie noted, “When we were like, ‘Why do weneed to know the meaning of this?’ and it’s so you can explain it, becauseyou can’t teach something to a student if you don’t know what it means”.

Many noted that the experience of the CMTE made the contents ofthe mathematics class more compelling. Phan noted, “We understand thatknowing math is easy but knowing how to teach math to children is hardwork”. Ana expanded on this observation:

I think that as students, we tend to think when we are learning these math concepts that it’sso obvious and easy and that it will be easy to teach them to students. But it’s not as easyas we think it is, and the students tend not to know as much as we think they know. We’renot just going to go in and knock them dead. It’s going to take work and thought.

The prospective teachers began to distinguish doing mathematics, whichthey equated with using memorized procedures and considered easy,from teaching mathematics, which they equated with understanding andconsidered more difficult.

Many shared Ana’s observation that the children did not know as muchas the prospective teachers had expected them to and remarked that withoutthe CMTE they would have doubted their instructor when he told themthat children have trouble with particular concepts or tend to think aboutproblems in specific ways. Gloria observed,

The instructor would have said, “This is how the kids are learning”, but that wouldn’t havemeant anything to me. I would have been like, “So . . .? Okay. I’ll just concentrate on doing


the math problem”. But during the CMTE, it’s like you’re right there and applying whatyou’re learning and seeing whether or not it works.

When the prospective teachers had first-hand experiences teaching math-ematics, they realized that they needed to understand it well and beganto appreciate the value of their mathematics class. As Gloria pointedout, without the practical experience, most would have focused only onmastering techniques for solving problems instead of on the mathematicalconcepts related to those techniques.

Beliefs about Children’s Informal Knowledge

The first three sessions of the CMTE, those with the 6–9-year-old children,were intended to acquaint the prospective teachers with the informalknowledge that children bring to school. By having the prospectiveteachers pose story problems to young children, we had hoped that theprospective teachers would see that the children could model the actionof the problem using manipulatives and solve a variety of story problemswithout much guidance. Most of the children that were interviewed haddifficulty with several of the story problems, perhaps because they hadnot had any exposure to using manipulatives to model story problems intheir classrooms. Although many children can readily solve such problems(Carpenter, Fennema, Franke, Levi & Empson, 1999), the children in theCMTE could not. They tended simply to add the two numbers in the prob-lems posed, regardless of the action in the problem. Although many ofthe children could solve multiplication and division problems set in storycontexts, they were so unsuccessful with some of the other problem types(comparison situations, join change unknown, etc.) that the prospectiveteachers tended to focus on the children’s difficulties rather than on theirsuccesses.

The prospective teachers found the interviews with the primary-gradechildren to be awkward because they saw each child only once and didnot have opportunities to develop rapport with the children. They also haddifficulty observing while their child was working instead of showing thechild how to solve problems. Holly noted, “It was challenging for me tohold back and not help him by hinting at ways to solve the problems he washaving difficulty with”. The children struggled to explain their thinking,probably because explanations were not asked of them in their classrooms.Ana noted, “The primary interviews didn’t make much of an impressionon me, because I didn’t expect much out of the kids, to be perfectlyhonest, and we didn’t get much”. Unfortunately, the work with the primarychildren left little impression on most of the prospective teachers, and itwas unclear how their work with the primary children affected their beliefs.

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BUILDING ON EXISTING BELIEF SYSTEMS

The belief-system change that I have attempted to illustrate focused onthe elaboration of attitudes through the development of new beliefs andthe formation of new connections among beliefs due to an intense exper-ience coupled with reflection. We speculate that the prospective teachers’original attitudes about teaching were undifferentiated, consisting of fewbeliefs, because of their limited experiences. Inasmuch as they had not hadopportunities to reflect on their beliefs about teaching, the beliefs remainedat the subconscious level. When they struggled to teach fractions to the 10-year-olds, their beliefs about teaching became more salient to them. Theyadded to their beliefs the notions that teachers should listen to children todetermine at what point to begin instruction, provide children with timeto think, and be prepared to explain concepts in a variety of ways. This isan example of the elaboration of an attitude when new beliefs emerge andbecome connected to existing beliefs. The basis of this belief change wasan intense experience.

The struggles that many of the prospective teachers had, when theyattempted to help their children understand fractions better, could becharacterized as failed teaching experiences. Weinstein (1990) suggestedthat failed teaching experiences were critical in helping prospectiveteachers to overcome their optimistic bias about their abilities as teachers.Weinstein speculated that prospective teachers need to see that teaching isnot as easy as they had believed and that facing the challenge of teachingstudents, particularly those who struggle, would affect their beliefs. Thisseems to be the case for the prospective teachers in the CMTE. As Lisawrote, “I am beginning to find out that teaching is not as easy as is looks.It takes a lot more to be a teacher than enjoying working with children”.The prospective teachers’ CMTE experiences helped them to recognizethat mathematics teaching is much more complicated than they expectedit to be and helped them to appreciate the value of the material in themathematics course.

Note that most of the prospective teachers continued to hold ontotheir beliefs that teaching involves explaining things to children, eventhough they spoke of the importance of giving children time to think forthemselves. This was evident in the prospective teachers’ actions in theproblem solving sessions when they presented children with the standardalgorithm for fraction addition. We concluded from their actions that, for atleast the six prospective teachers in these three partnerships, their actionsindicated a “teaching as telling” belief along with a belief about mathe-matics learning as the acquisition of standard symbolic procedures. Wedid not conclude that our efforts to help them change their beliefs werewasted. Instead we interpreted these examples as evidence that prospective


teachers do not let go of old beliefs while they are forming new ones. Theirnew beliefs about the importance of multiple solution strategies, or theirknowledge of these approaches, or both of these, were not strong enoughto compel them to introduce multiple approaches for adding fractions. Wewere encouraged that the prospective teachers could be critical of theirown actions, and we concluded that while their beliefs grew, several newbeliefs which might compel them to make different teaching decisions inthe future, would be added to their belief systems. Their lack of success inteaching the addition of fractions may accelerate their acquisition of newbeliefs while they experience the limitation of their existing belief system.The mismatch between the prospective teachers’ comments in their inter-views and surveys and their decisions with their students may be construed,not as evidence of conflicting beliefs, but as evidence of evolving beliefs.Although belief change had been initiated, the IMAP team recognized thatthe prospective teachers had not developed all the beliefs we would likethem eventually to develop. Over time, the prospective teachers’ beliefsystems may continue to change when they have more experiences uponwhich they have opportunities to reflect. More beliefs may be added totheir belief systems, and their beliefs about teaching-as-telling may stillexist but may be weaker or less central.

Providing prospective teachers with intense experiences that involvethem intimately with children poses a promising avenue for belief change.Coupling these experiences with reflection allows the beliefs that arisefrom these experiences to be examined and refined. The CMTE cameearly in the teacher preparation program. Given the incremental natureof belief change, teacher educators might consider creating several suchexperiences throughout the teacher preparation program, especially whileprospective teachers are doing their subject matter preparation, to ensurethat the beliefs become well connected and that attitudes become differen-tiated. One intense experience is a starting point but seems insufficient tocatalyze all the belief change that teacher educators might desire. Variousexperiences are required to help prospective teachers develop new beliefsabout mathematics, and teacher educators would be wise to recognize thatthese beliefs will probably coexist with, rather than replace, the beliefs thatpreceded them.

ACKNOWLEDGEMENTS

This work was supported through the Interagency Educational ResearchInitiative (IERI) Grant G00002211. Any opinions expressed herein arethose of the author and do not necessarily reflect the views of IERI.

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The author wishes to thank her colleagues, Bonnie Schappelle, RandyPhilipp, Lisa Clement, Jennifer Chauvot, Cheryl Vincent, and JudySowder, for their help with this article.

REFERENCES

Ambrose, R. (2002, April). They all hate math: Getting beyond our stereotypes ofprospective elementary school teachers. Paper presented at the annual meeting of theAmerican Educational Research Association, New Orleans.

Ball, D. (1990). Prospective elementary and secondary teachers’ understanding of division.Journal for Research in Mathematics Education, 21, 132–144.

Ball, D. (2000). Bridging practices: Intertwining content and pedagogy in teaching andlearning to teach. Journal of Teacher Education, 51, 241–247.

Carpenter, T.P., Fennema, E., Franke, M., Levi, L. & Empson, S. (1999). Children’smathematics: Cognitively guided instruction. Portsmouth, NH: Heinemann.

Conference Board of Mathematical Sciences (2001). The mathematical educationof teachers. Washington, DC: Mathematical Association of America. http://www.cbmsweb.org/MET_Document/index.htm

Feiman-Nemser, S. & Buchmann, M. (1986). The first year of teacher preparation:Transition to pedagogical thinking. Journal of Curriculum Studies, 18, 239–256.

Feiman-Nemser, S., McDiarmid, G.W., Melnick, S. & Parker, M. (1988). Changing begin-ning teachers’ conceptions: A study of an introductory teacher education course. EastLansing: National Center for Research on Teacher Education and Department of TeacherEducation, Michigan State University.

Fenstermacher, G. (1979). A philosophical consideration of recent research on teachereducation. In L.S. Shulman (Ed.), Review of research in education (Vol. 6, pp. 157–185).Itasca, IL: F.E. Peacock.

Gellert, U. (2000). Mathematics instruction in safe space: Prospective elementary teachers’views of mathematics education. Journal of Mathematics Teacher Education, 3, 251–270.

Glaser, B. (1998). Doing grounded theory: Issues and discussions. Mill Valley, CA:Sociology Press.

Goldstein, L. (1999). The relational zone: The role of caring relationships in the co-construction of mind. American Educational Research Journal, 36, 647–673.

Goodman, J. (1988). Constructing a practical philosophy of teaching: A study ofprospective teachers’ professional perspectives. Teaching and Teacher Education, 4,121–137.

Green, T.F. (1971). The activities of teaching. New York: McGraw-Hill.Hargreaves, A. (1994). Changing teachers, changing times. London: Cassell.Hollingsworth, S. (1989). Prior beliefs and cognitive change in learning to teach. American

Educational Research Journal, 26, 160–189.Hollingsworth, S. (1992). Learning to teach through collaborative conversation: A feminist

approach. American Educational Research Journal, 29, 373–404.Howes, E.V. (2002). Learning to teach science for all in the elementary grades: What do

preservice teachers bring? Journal of Research in Science teaching, 39, 845–869.Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understanding

of fundamental mathematics in China and the United States. Mahwah, NJ: Erlbaum.


Mack, N. (1995). Confounding whole-number and fraction concepts when building oninformal knowledge. Journal for Research in Mathematics Education, 26, 422–443.

Mathematical Sciences Education Board (2001). Knowing and learning mathematics forteaching: Proceedings of a workshop. Washington, DC: National Academy Press.

McDiarmid, G.W., Ball, D.L. & Anderson, C.W. (1989). Why staying one chapter aheaddoesn’t really work: Subject specific pedagogy. In M.C. Reynolds (Ed.), Knowledge basefor the beginning teacher (pp. 193–206). Oxford: Pergamon Press.

McLaughlin, H.J. (1991). Reconciling care and control: Authority in classroom relation-ships. Journal of Teacher Education, 42(3), 182–195.

Nespor, J. (1987). The role of beliefs in the practice of teaching. Journal of CurriculumStudies, 19, 317–328.

Pajares, M.F. (1992). Teachers’ beliefs and educational research: Cleaning up a messyconstruct. Review of Educational Research, 62, 307–332.

Richardson, V. (1996). The role of attitudes and beliefs in learning to teach. In J. Sikula(Ed.), Handbook of research on teacher education (pp. 102–119). New York: Macmillan.

Rokeach, M. (1968). Beliefs, attitudes, and values: A theory of organization and change.San Francisco: Jossey-Bass.

Simon, M., Tzur, R., Heinz, K., Kinzel, M. & Smith, M.S. (2000). Characterizing aperspective underlying the practice of mathematics teachers in transition. Journal forResearch in Mathematics Education, 31, 579–601.

Stofflett, R. & Stoddart, T. (1992, April). Patterns of assimilation and accommodationin traditional and conceptual change teacher education course. Paper presented at theAnnual Meeting of the American Educational Research Association, San Francisco.

Weinstein, C.S. (1989). Teacher education students’ preconceptions of teaching. Journalof Teacher Education, 4, 31–40.

Weinstein, C.S. (1990). Prospective elementary teachers’ beliefs about teaching: Implica-tions for teacher education. Teaching and Teacher Education, 6, 279–290.

Wideen, M., Mayer-Smith, J. & Moon, B. (1998). A critical analysis of the research onlearning to teach: Making the case for an ecological perspective on inquiry. Review ofEducational Research, 68(2), 130–178.

Worthy, J. & Patterson, E. (2001). “I can’t wait to see Carlos!”: Situated learning andpersonal relationships with students. Journal of Literacy Research, 33(2), 303–344.

University of California-DavisOne Shields AvenueDavis, CA 95616USAE-mail: [email protected]

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Initiating Change in Prospective Elementary School Teachers' Orientations to Mathematics Teaching by Building on Beliefs